Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 63 / 5 / 3 / Solution

The $[1/1]$ Padé approximant of the exponential is $r(z)=(1+z/2)/(1-z/2)$. Replacing each exponential in its original order gives the split Crank-Nicolson methodwith the analogous definition for $B$. Operationally, first solve the $B$ substep and then the $A$ substep. The matrices $I-\delta A/2$, $I-\delta B/2$ are invertible for every $\delta \ge 0$, since $A,B$ are symmetric negative definite.

An orthonormal eigenbasis of $A$ diagonalizes $C_A$. Its eigenvalue for $\lambda \le 0$ isTherefore $\Vert C_A{\Vert }_h\le 1$, and similarly $\Vert C_B{\Vert }_h\le 1$. Submultiplicativity gives the contractivity of split Crank-Nicolson diffusion:No commutativity or simultaneous diagonalization is needed for this product bound. This proves unconditional discrete $L^2$ stability, not positivity of every stencil weight or unconditional maximum-norm monotonicity. Nor do the individually second-order rational substeps make the ordered splitting second order: when $[A,B]\ne 0$, its leading splitting defect remains ${\delta }^2[A,B]/2$.